Relation between bulk order-parameter correlation function and finite-size scaling

Abstract

We study the large-distance behavior of the bulk order-parameter correlation function G(r) for T > T c within the lattice version of the ϕ 4 theory including lattice effects. We also study the large-L behavior of the susceptibility χ for T > T c of the confined lattice system of linear size L with periodic boundary conditions. We find that the structure of the large-L behavior of χ of the confined system is closely related to the structure of the large-distance behavior of G(r) of the bulk system. Explicit results are derived in the spherical (large-n) limit and in oneloop order for general dimensions d > 2. For the lattice model with cubic symmetry we find that finite-size scaling must be formulated in terms of the anisotropic bulk correlation length (exponential correlation length) that governs the exponential decay of G(r) for large r rather than in terms of the ordinary isotropic bulk correlation length ξ defined via the second moment of G(r). We show that it is the exponential bulk correlation length ξ 1 in the direction of the cubic axes is not captured by the standard perturbation approach that separates the homogeneous lowest mode from the inhomogeneous higher modes.Consequences for the theory of finite-size effects above four dimensions are discussed. We show that the two-variable finite-size scaling form predicts an exponential approach ∝ e −L/ξ 1 to the bulk critical behavior above T c whereas the reduction to a single-variable scaling form implies a power-law approach ∝ L −d .

Crossover from low-temperature to high-temperature fluctuations: Universal and nonuniversal Casimir forces of isotropic and anisotropic systems

“…III. C) in the direction of one of the cubic axes [40,67,68] ξ e = (ã/2) {arsinh [ã/(2ξ + )]} −1 must be employed. (The latter expression is a one-loop result for finite n and is exact for n → ∞ [68].)…”

“…C) in the direction of one of the cubic axes [40,67,68] ξ e = (ã/2) {arsinh [ã/(2ξ + )]} −1 must be employed. (The latter expression is a one-loop result for finite n and is exact for n → ∞ [68].) For the case of a block geometry and for a nearest-neighbor interaction, this regime is characterized by L α 24(ξ + ) 3 /ã 2 , α = 1, ..., d. Similar remarks apply to the exponential tail of X − , (5.16), for n = 1 below T c [23], and to the scaling functions above T c in film geometry (Sec.…”

“…In (2.67) and (2.68) the asymptotic amplitudes of the "true" correlation lengths are employed as defined in [23] through the exponentially decaying part of the correlation function [23,[79][80][81]. We conjecture that the ratio of these amplitudes is identical with the ratio of the corresponding second-moment principal correlation lengths of the anisotropic Ising model.…”

“…2 of [40], shaded region) where the exponential decay of G b is governed by a "true" or "exponential" correlation length ξ e± (t) = ξ ± (t) [40,60,[67][68][69]. The ratio lim t→0 ξ e± (t)/ξ ± (t) is universal [60,69] and equal to 1 for n = ∞ [68].…”

Universality of the thermodynamic Casimir effect

“…In particular one has r * ∼ ξ 2 for the Ising model or within mean-field theory, r * ∼ ξ 3 for the spherical model [40], and r * ∼ ξ log ξ for subleading long-ranged interactions [37,46]. This view has recently been challenged by Chen and Dohm [18] who purportedly report, for systems with short-ranged interactions, leading finite-size contributions different from the ones expected from the above discussion.…”